Global Hopf bifurcation of two-parameter flows
The author analyzes the behavior of center-indices introduced by \textit{J. Mallet-Paret} and \textit{J. Yorke} [J. Differ. Eq. 43, 412-450 (1982; Zbl 0487.34038)] for two parameter flows generated by an equation \(\dot x=f(\lambda,x)\), \(\lambda \in {\mathbb{R}}^ 2\). The integer sum of center- indices along a one-dimensional curve in parameter space is called the H- index. A nonzero H-index implies global Hopf bifurcation. The index H is not a homotopy invariant because of the occurrence of stationary points \(F^ 1(0)\) where the differential D F has an algebraically double eigenvalue zero, which are called B-points. To each B-point the author assigns an integer B-index which relates to the H-index by a formula similar to one in the calculus of residues. The formula is applied to study global bifurcations of periodic solutions in chemical problems.
scientific article; zbMATH DE number 3973886
scientific article